Vaughan and Torkel have both given me something to think about, to clear up
my position. (Incidentally, I have been misspelling Vaughan's name leaving
out the second "a", and people have been misspelling my last name, putting
in two "c"s.)
From: torkel at sm.luth.se (Torkel Franzen)
Date: Wed, 19 Nov
Torkel writes:
> Colin McLarty says:
>>Similarly, categorical foundations do not begin with a class of
>>objects and one of arrows. They begin by positing arrows and a
>>composition relation, and stating assumptions about them.
> You speak of "categorical foundations", whereas Feferman's comments
refer >to "foundations of category theory". This perhaps reflects a
difference in >what the two of you have in mind.
I refer to categorical foundations of category theory, among other
subjects. And these are the main subject of Feferman's 1977 paper.
> I don't think anybody will dispute that category theory can be
formulated >by positing arrows and a composition relation, and stating
assumptions about >them.
Yes, Kreisel and Feferman and others insist on this point. And yet
they also say things like:
>> Basically, the argument is that the
>> notion of category is defined in terms of the notions of collection and
>>operation, viz. the collection of all the "objects" of the category, the
>>collection of all its "morphisms", and the operations of domain,
>>co-domain, and composition applied to morphisms and of 1_a applied to
>>objects a. (Alternatively, one can deal with just the collection of
>>morphisms alone.). Thus the foundations of category theory must be given
>>in terms of some sort of theory of operations (or functions) and
>>collections (or classes). (Feferman's post of Nov.17)
This claim seems quite groundless to me. It seems to be refuted by
many widely published examples, some of which I mentioned in my post. In
fact, it seems to me SO ENTIRELY wrong that I think I may misunderstand it
in some fundamental way. I would like to see a defense of this claim.
I claim that category theory (and many other subjects including set
theory, vector spaces, differential geometry, and probably recursive
analysis though this is an opem problem now) can be axiomatized by first
order extensions of the axioms I gave in my post--and thus given logical
foundations independent of any prior theory of operations and collections.
Torkel goes on to the more philosophic question of whether such
axiomatizations can be considered foundations in an epistomological sense:
> But now suppose I ask about the meaning and justification of the
basic >theory of categories. The answer I'm used to is (essentially) that
these axioms >sum up, in a way that turns out be very useful, basic aspects
and properties of >structures in different parts of mathematics. By looking
at a number of >examples, I get to understand how this might be so. Mappings
of various kinds, >I find, seem to be the basic source of inspiration for
and instantiation of the >categorical axioms.
> Therefore, even if theories of those other basic concepts can be cast
in >categorical terms, categories seem to be secondary in terms of our
>understanding of mathematics.
He says that axioms of set theory too are meaningful to us only
after we are acquainted with informal mathematical examples:
>Still, in presenting the
>foundations of set theory, we invoke a concept of "a bunch of things" which
is >not abstracted from various mathematical instantiations of that concept,
but is >understood and pictured as a primitive notion.
...
> In category theory, as I understand it, there is no similar primitive
>notion to be had.
Oh yes, I think there is. I think the notion of "operation" (including
"symmetry" and "growth" and more as special cases) is just as primitive as
the notion of "collection". I take it that Feferman agrees with me on this.
Where he
disagrees with me is when I say the category axioms (which I gave as C1-3)
mathematize this primitive notion just as foundationally as set theory
mathematizes collections.
Vaughan's summary of my position seems accurate to me. But he raises two
concerns:
From: pratt at cs.Stanford.EDU (Vaughan R. Pratt)
Date: Tue, 18 Nov wrote:
> First, I'd like to know how a category theorist would interpret the
>first-order predicate "x is incident on y" (i.e. "object y is either the
source >or target of arrow x"). To a set theorist, this predicate
associates to each >object y a class of arrows. In the absence of a notion
of class, what >interpretation does a category theorist place on it?
Certainly not a >subcategory of the universe, since for any given y this
class need not be >closed under composition. And there is no evident
alternative structure for >this class, which looks awfully like
>an unstructured class.
Of course in first order set theoretic or category theoretic foundations
for category theory we do not "interpret" this relation at all, in the sense
of providing an entity as its extension. I take it that Vaughan sees this.
His concern is, if we choose to ascend to a higher level so that extensions
become actual entities, how can that level usefully be anything other than
the notion of class? That is, how can it be anything naturally categorical?
Actually, in practice we rarely want the unstructured class of arrows to a
fixed object y. Rather we want the category with those arrows as objects,
and commutative triangles over y as arrows, that is the slice category over
y. We often want to know if this slice category is large or small, and in
many situations this can equivalently be phrased as asking whether or not
the class of arrows forms a set. But we work with the slice category and not
the class of arrows--and in fact the answer is generally no, the category is
not small, but there is a small category cofinal in it which will serve out
purposes. That answer requires the category structure of the slice, not just
the class of arrows. (For typical examples, see Grothendieck et al.
SEMINAIRE DE GEOMETRIE ALGEBRIQUE especially vol.4, or Tamme, INTRODUCTION
TO ETALE COHOMOLOGY.)
> Second, there is the question of whether Cat is in fact a category as I
>think Colin is implying, or should more properly be understood as a
2-category.
Yes, for a lot of purposes the 2-category of categories, functors, and
natural transformations is more important then the 1-category of categories
and functors. And I wonder what insights would come from axiomatizing the
2-category. But I think the basic question of whether there can be
categorical foundations for category theory is already well approached if we
get the 1-category.
Colin McLarty